View-obstruction problems. ii

Let Sⁿ denote the region 0 < x_i < ∞ (i = 1, 2, …, n) of n-dimensional Euclidean space Eⁿ. Suppose C is a closed convex body in Eⁿ which contains the origin as an interior point. Define αC for each real number α ≥ 0 to be the magnification of C by the factor a and define C + (m₁, …, m_n) for each point (m₁, …, m_n) in Eⁿ to be the translation of C by the vector (m₁, …, m_n). Define the point set Δ(C, α) by Δ(C, α) = {αC + (m₁ + ½, …, m_n + ½): M₁, …, m_n nonnegative integers}. The view-obstruction problem for C is the problem of finding the constant K(C) defined to be the lower bound of those a such that any half-line L given by x_i = a_it (i = 1, 2, …, n), where the a_i (1 ≤ i ≤ n) are positive real numbers, and the parameter t runs through [0, ∞), intersects Δ(C, α). The paper considers the case where C is the n-dimensional cube with side 1, and in this case the constant K(C) is known for n ≤ 3. The paper gives a new proof for the case n = 3. Unlike earlier proofs, this one could be extended to study the cases with n ≥ 4.